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Real Variable Inverse Laplace Transform

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Analysis as a Life

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Abstract

The aim of this work is to provide a review of authors’ contributions to the field of the Laplace transform in the last 20 years.

Dedicated to the 80th anniversary of Prof. Heinrich Begehr

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References

  1. J. Abate, W. Whitt, The Fourier-series method for inverting transforms of probability distributions. Queueing Syst. 10, 5–88 (1992)

    Article  MathSciNet  Google Scholar 

  2. L. Amério, Sulla transformata doppia di Laplace. Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. 12, 707–780 (1941)

    MATH  Google Scholar 

  3. R. Bellman, R.E. Kabala, R.A. Lockett, Numerical Inversion of the Laplace Transform (Elsevier, New York, 1966)

    Google Scholar 

  4. A. Boumenir, A. Al-Shuaibi, The inverse Laplace transform and analytic pseudo-differential operators. J. Math. Anal. Appl. 228(1), 16–36 (1998)

    Article  MathSciNet  Google Scholar 

  5. A. Boumenir, A. Al-Shuaibi, On the numerical inversion of the Laplace transform by the use of optimized Legendre polynomials. Approx. Theory Appl. 16(4), 17–32 (2000)

    MathSciNet  MATH  Google Scholar 

  6. A. Boumenir, V.K. Tuan, Sampling eigenvalues in Hardy spaces. SIAM J. Numer. Anal. 45(2), 473–483 (2007)

    Article  MathSciNet  Google Scholar 

  7. A. Boumenir, V.K. Tuan, The computation of eigenvalues of singular Sturm-Liouville operators. Adv. Appl. Math. 39, 222–236 (2007)

    Article  MathSciNet  Google Scholar 

  8. A. Boumenir, V.K. Tuan, The interpolation of the Titchmarsh-Weyl function. J. Math. Anal. Appl. 335, 72–78 (2007)

    Article  MathSciNet  Google Scholar 

  9. A. Boumenir, V.K. Tuan, Representation and sampling of Hardy functions. Math. Methods Appl. Sci. 33(4), 485–492 (2010)

    MathSciNet  MATH  Google Scholar 

  10. A. Boumenir, V.K. Tuan, An inverse problem for the heat equation. Proc. Am. Math. Soc. 138(11), 3911–3921 (2010)

    Article  Google Scholar 

  11. A. Boumenir, V.K. Tuan, Recovery of a heat equation by four measurements at one end. Numer. Funct. Anal. Optim. 31(2), 155–163 (2010)

    Article  MathSciNet  Google Scholar 

  12. L.P. Castro, H. Fujiwara, M.M. Rodrigues, S. Saitoh, A new discretization method by means of reproducing kernels, in Proceedings of the 20th International Conference on Finite or Infinite Dimensional Complex Analysis and Applications, Hanoi University of Science and Technology, Hanoi, July 29–August 3, 2012

    Google Scholar 

  13. A.H.-D. Cheng, P. Sidauruk, Y. Abousleiman, Approximate inversion of the Laplace transform. Math. J. 4(2), 76–82 (1994)

    Google Scholar 

  14. B. Davies, B. Martin, Numerical inversion of the Laplace transform: a survey and comparison of methods. J. Comput. Phys. 33, 1–32 (1979)

    Article  MathSciNet  Google Scholar 

  15. G. Doetsch, Handbuch der Laplace Transformation, vol. 1 (Birkhäuser, Basel, 1950)

    Book  Google Scholar 

  16. D.G. Duffy, On the numerical inversion of Laplace transforms: comparison of three new methods on characteristic problems from applications. ACM Trans. Math. Softw. 19(3), 333–359 (1993)

    Article  MathSciNet  Google Scholar 

  17. G. Honig, U. Hirdes, Algorithm 27: a method for the numerical inversion of Laplace transforms. J. Comput. Appl. Math. 10, 113–132 (1984)

    Article  MathSciNet  Google Scholar 

  18. V.I. Krylov, N.S. Skoblya, A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transform (Mir, Moscow, 1977)

    Google Scholar 

  19. B.M. Levitan, Inverse Sturm-Liouville Problems (VNU Science Press, Utrecht, 1987)

    Book  Google Scholar 

  20. A. Murli, M. Rizzardi, Algorithm 682: Talbot’s method for the Laplace inversion problem. AMS Trans. Math. Softw. 16(2), 158–168 (1990)

    Article  Google Scholar 

  21. R.E.A.C. Paley, N. Wiener, Fourier Transforms in the Complex Domain (Colloquium Publications American Mathematical Society, Providence, 1934)

    MATH  Google Scholar 

  22. J. Peng, S.K. Chung, Laplace transforms and generators of semigroups of operators. Proc. Am. Math. Soc. 126(8), 2407–2416 (1998)

    Article  MathSciNet  Google Scholar 

  23. R. Piessens, Algorithm 453: Gaussian quadrature formulas for Bromwich’s integral [D1]. Commun. ACM 6(8), 486–487 (1973)

    Article  Google Scholar 

  24. R. Piessens, R. Huysmans, Algorithm 619: automatic numerical inversion of the Laplace transform [D5]. ACM Trans. Math. Softw. 10, 348–353 (1984)

    Article  Google Scholar 

  25. E.L. Post, Generalized differentiation. Trans. Am. Math. Soc. 32, 723–781 (1930)

    Article  MathSciNet  Google Scholar 

  26. A.P. Prudnikov, Y.A. Brychkov, O.I. Marichev, Integrals and Series. Volume 4: Direct Laplace Transforms (Gordon and Breach, New York, 1992)

    Google Scholar 

  27. A. Rybkin, V.K. Tuan, A new interpolation formula for the Titchmarsch-Weyl m-function. Proc. Am. Math. Soc. 137(12), 4177–4185 (2009)

    Article  Google Scholar 

  28. H. Stehfest, Algorithm 368: numerical inversion of Laplace transforms [D5]. Commun. ACM 13(1), 47–49 (1970)

    Article  Google Scholar 

  29. V.K. Tuan, On the factorization of integral transformations of convolution type in the space \(L_{2}^{\Phi }\) (Russian). Dokl. Akad. Nauk. Armyan SSR 83(1), 7–10 (1986)

    Google Scholar 

  30. V.K. Tuan, Laplace transform of functions with bounded averages. Int. J. Evol. Equ. 1(4), 429–433 (2005)

    MathSciNet  MATH  Google Scholar 

  31. V.K. Tuan, A. Boumenir, Sampling in Paley-Wiener and Hardy spaces, §9 in Harmonic, Wavelet and p-adic Analysis, ed. by N.M. Chuong, Y.V. Egorov, M.Y. Khrennikov, D. Mumford (World Scientific Publishing, Hackensack, 2007), pp. 175–209

    Google Scholar 

  32. V.K. Tuan, D.T. Duc, Automatic evaluation of abscissa of convergence for inverse Laplace transform. Frac. Calc. Appl. Anal. 3(4), 353–358 (2000)

    MathSciNet  MATH  Google Scholar 

  33. V.K. Tuan, D.T. Duc, Convergence rate of Post-Widder approximate inversion of the Laplace transform. Vietnam J. Math. 28(1), 93–96 (2000)

    MathSciNet  MATH  Google Scholar 

  34. V.K. Tuan, D.T. Duc, A new real inversion formula for the Laplace transform and its convergence rate. Frac. Cal. Appl. Anal. 5(4), 387–394 (2002)

    MathSciNet  MATH  Google Scholar 

  35. V.K. Tuan, N.T. Hong, Interpolation in the Hardy space. Integr. Transf. Spec. Funct. 24(8), 664–671 (2013)

    Article  MathSciNet  Google Scholar 

  36. V.K. Tuan, T. Tuan, A real-variable inverse formula for the Laplace transform. Integr. Transf. Spec. Funct. 23(8), 551–555 (2012)

    Article  MathSciNet  Google Scholar 

  37. V.K. Tuan, O.I. Marichev, S.B. Yakubovich, Composition structure of integral transformations. Soviet Math. Dokl. 33(1), 166–170 (1986)

    MATH  Google Scholar 

  38. F. Veillon, Algorithm 486: numerical inversion of Laplace transform [D5]. Commun. ACM 17(10), 587–589 (1974)

    Article  Google Scholar 

  39. D.V. Widder, The inversion of the Laplace integral and the related moment problem. Trans. Am. Math. Soc. 36, 107–200 (1934)

    Article  MathSciNet  Google Scholar 

  40. D.V. Widder, The Laplace Transform (Princeton University Press, Princeton, 1946)

    Google Scholar 

  41. S.B. Yakubovich, A real inversion formula for the bilateral Laplace transform (Russian). Izv. Nats. Akad. Nauk Armenii Mat. 40(3), 67–79 (2005)

    MathSciNet  Google Scholar 

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Acknowledgement

The work of the third author was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2017.310.

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Correspondence to Vu Kim Tuan .

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Tuan, V.K., Boumenir, A., Duc, D.T. (2019). Real Variable Inverse Laplace Transform. In: Rogosin, S., Çelebi, A. (eds) Analysis as a Life. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-02650-9_15

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